بسم الله الرحمن الرحيم
(1) It is forgiven to eat unknowingly during fast. Hence if your mother drank water in the state of fasting while she did not remember the fast then her fast shall be valid and it will not break. However, if she asks someone for water and he remembers about her fast then he should remind her of the fast.
(2) Mere doubt shall not be taken into consideration. The ruling is the same which was mentioned above. However, while the fast was remembered and water went down the throat while doing gargling then the fast shall break but only qaza shall be wajib not kaffarah. Kaffarah becomes wajib only when a man eats something during the fast of Ramadan intentionally. And if someone remembers his fast or not but he eats something by mistake then kaffarah does not become wajib. Yes, only qaza shall be wajib in latter case.
Allah knows Best!
Darul Uloom Deoband